Advanced Futoshiki Techniques for Harder Grids

The advanced Futoshiki techniques are three ways of using signs more strongly than one pair at a time. They matter on the hard (7 x 7) and expert (8 x 8) grids of LogikLand, where there are few given numbers and many signs. We follow one real hard grid from its first scan to the end, so every step can be checked. If you are new to the puzzle, begin with how to solve Futoshiki.

What changes on 7 x 7 and 8 x 8

In the grids we generated, hard puzzles had 0 to 5 given numbers and 18 to 30 signs, and expert puzzles had 0 to 7 given numbers and 24 to 43 signs (14 grids of each level traced, plus 60 grids measured for the counts). The signs form long chains: the longest chain was 3 to 6 squares on hard and 4 to 7 on expert. And the most frequent way a square gets its number is a number that has only one place left in a row or column: that was the commonest reason in 11 of 14 hard grids and in 11 of 14 expert grids. These three facts explain the techniques below.

The grid we use

This hard grid has one given number (a 6 in the top row) and 25 signs. Rows are counted from the top and columns from the left, starting at 1. The solution is on the right.

Technique 1: chain windows

If k squares form a chain a1 < a2 < ... < ak in an n x n grid, then the i-th square must be at least i and at most n - k + i. We call that range its window. A window is n - k + 1 numbers wide, so the longer the chain, the tighter the window.

This grid has a chain of six squares in a 7 x 7 grid, running around a block of three rows and two columns. In order from the smallest square: row 3 column 6, row 4 column 6, row 4 column 7, row 3 column 7, row 2 column 7 and row 2 column 6. Here n - k + 1 = 2, so each window has two numbers.

Write each window in the corner of its square, for example "1-2". With no given number, you have already limited six squares.

Technique 2: hunting the biggest number

A square that is smaller than one of its neighbours can never hold the biggest number, 7. Look down a column or along a row and cross off every square that is smaller than a neighbour. If one square is left, the 7 goes there.

In column 2 of this grid, rows 1, 3, 4, 5, 6 and 7 are each smaller than a neighbour: row 1 is smaller than row 2, row 3 is smaller than row 2, row 4 is smaller than row 3, and so on. Only the square at row 2 is never the smaller one. So the 7 of column 2 sits at row 2, column 2. The same idea works for 1: a square that is bigger than a neighbour can never be 1.

Technique 3: one anchor, a whole chain

The windows are not enough by themselves, because each square still has two possibilities. A single sure number settles the whole chain, as a row of dominoes:

1. Row 2 already holds the 7 (technique 2). So the square at row 2, column 6 cannot be 7: its window leaves 6.
2. The square at row 2, column 7 is smaller than it, and its window is 5 or 6: it is 5.
3. The next square (row 3, column 7) is smaller than 5, window 4 or 5: it is 4.
4. Row 4, column 7 is smaller than 4, window 3 or 4: it is 3.
5. Row 4, column 6 is smaller than 3, window 2 or 3: it is 2.
6. Row 3, column 6 is smaller than 2, window 1 or 2: it is 1.

Six numbers are placed from one anchor. You can check them against the solution: the chain reads 1, 2, 3, 4, 5, 6.

Keeping notes on a big grid

On a 7 x 7 or 8 x 8 grid, writing every candidate is slow. Our suggestion, not a rule of the puzzle: write only windows and the numbers you are sure of. Mark a "7" or a "1" hunt in the margin of each row and column: "7?" next to a column means you have not yet looked for where its biggest number goes. The generator's solver uses nothing else than the line logic, the signs and these chains, so you never need a trick beyond them. If you know others, such as pairs of squares with the same two candidates, you may use them, but a LogikLand grid never requires them.

Practice

Train the windows on small grids with the printable easy Futoshiki puzzles, then move up. The hard and expert grids are made by the Futoshiki assistant once you create a free account to make a full book. For the rules in one page, see Futoshiki rules.